Margherita Disertori

Vanishing of Beta Function of Non Commutative Phi**4(4) Theory to all orders

Abstract The simplest non-commutative renormalizable field theory, the ϕ 4 model on four-dimensional Moyal space with harmonic potential is asymptotically safe up to three loops, as shown by H. Grosse and R. Wulkenhaar, M. Disertori and V. Rivasseau. We extend this result to all orders.

Two- and three-loop beta function of non-commutative Φ4 4 theory

The simplest non-commutative renormalizable field theory, the φ44 model on four dimensional Moyal space with harmonic potential, is asymptotically safe at one loop, as shown by Grosse and Wulkenhaar. We extend this result up to three loops. If this remains true at any loop, it should allow for a full non-perturbative construction of this model.

Density of states for random band matrices

Abstract: By applying the supersymmetric approach we rigorously prove smoothness of the averaged density of states for a three dimensional random band matrix ensemble, in the limit of infinite volume and fixed band width. We also prove that the resulting expression for the density of states coincides with the Wigner semicircle with a precision 1/W2, for W large but fixed.

Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model

We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.

Continuous Constructive Fermionic Renormalization

Abstract. We build the two dimensional Gross-Neveu model by a new method which requires neither cluster expansion nor discretization of phase-space. It simply reorganizes the perturbative series in terms of trees. With this method we can define non perturbatively the renormalization group differential equations of the model and at the same time construct explicitly their solution.

Interacting Fermi Liquid in Two Dimensions¶at Finite Temperature.¶Part I: Convergent Attributions

Abstract: Using the method of a continuous renormalization group around the Fermi surface, we prove that a two-dimensional interacting system of Fermions at low temperature T is a Fermi liquid in the domain , where K is some numerical constant. According to [S1], this means that it is analytic in the coupling constant λ, and that the first and second derivatives of the self energy obey uniform bounds in that range. This is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermion systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof.

Interacting Fermi Liquid in Two Dimensions at Finite Temperature. Part II: Renormalization

Abstract: This is a companion paper to [DR1]. Using the method of continuous renormalization group around the Fermi surface and the results of [DR1], we achieve the proof that a two-dimensional jellium system of interacting Fermions at low temperature T is a Fermi liquid above the BCS temperature. Following [S], this means proving analyticity in the coupling constant λ for , where K is some numerical constant, and some uniform bounds on the derivatives of the self-energy.

Interacting Fermi Liquid in Three Dimensions at Finite Temperature: Part I: Convergent Contributions

Abstract: Using the method of a continuous renormalization group around the Fermi surface, we prove that a two-dimensional interacting system of Fermions at low temperature T is a Fermi liquid in the domain , where K is some numerical constant. According to [S1], this means that it is analytic in the coupling constant λ, and that the first and second derivatives of the self energy obey uniform bounds in that range. This is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermion systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof.

Localization for a Nonlinear Sigma Model in a Strip Related to Vertex Reinforced Jump Processes

We study a lattice sigma model which is expected to reflect Anderson localization and delocalization transition for real symmetric band matrices in 3D, but describes the mixing measure for a vertex reinforced jump process too. For this model we prove exponential localization at any temperature in a strip, and more generally in any quasi-one dimensional graph, with pinning (mass) at only one site. The proof uses a Mermin–Wagner type argument and a transfer operator approach.

Density of States for Random Band Matrices in Two Dimensions

We consider a two-dimensional random band matrix ensemble, in the limit of infinite volume and fixed but large band width W. For this model, we rigorously prove smoothness of the averaged density of states. We also prove that the resulting expression coincides with Wigner’s semicircle law with a precision